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Mirrors > Home > MPE Home > Th. List > Mathboxes > eldm1cossres2 | Structured version Visualization version GIF version |
Description: Elementhood in the domain of restricted cosets. (Contributed by Peter Mazsa, 30-Dec-2018.) |
Ref | Expression |
---|---|
eldm1cossres2 | ⊢ (𝐵 ∈ 𝑉 → (𝐵 ∈ dom ≀ (𝑅 ↾ 𝐴) ↔ ∃𝑥 ∈ 𝐴 𝐵 ∈ [𝑥]𝑅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eldm1cossres 36572 | . 2 ⊢ (𝐵 ∈ 𝑉 → (𝐵 ∈ dom ≀ (𝑅 ↾ 𝐴) ↔ ∃𝑥 ∈ 𝐴 𝑥𝑅𝐵)) | |
2 | elecALTV 36399 | . . . 4 ⊢ ((𝑥 ∈ V ∧ 𝐵 ∈ 𝑉) → (𝐵 ∈ [𝑥]𝑅 ↔ 𝑥𝑅𝐵)) | |
3 | 2 | el2v1 36364 | . . 3 ⊢ (𝐵 ∈ 𝑉 → (𝐵 ∈ [𝑥]𝑅 ↔ 𝑥𝑅𝐵)) |
4 | 3 | rexbidv 3228 | . 2 ⊢ (𝐵 ∈ 𝑉 → (∃𝑥 ∈ 𝐴 𝐵 ∈ [𝑥]𝑅 ↔ ∃𝑥 ∈ 𝐴 𝑥𝑅𝐵)) |
5 | 1, 4 | bitr4d 281 | 1 ⊢ (𝐵 ∈ 𝑉 → (𝐵 ∈ dom ≀ (𝑅 ↾ 𝐴) ↔ ∃𝑥 ∈ 𝐴 𝐵 ∈ [𝑥]𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∈ wcel 2110 ∃wrex 3067 Vcvv 3431 class class class wbr 5079 dom cdm 5589 ↾ cres 5591 [cec 8477 ≀ ccoss 36327 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-sep 5227 ax-nul 5234 ax-pr 5356 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-clab 2718 df-cleq 2732 df-clel 2818 df-ral 3071 df-rex 3072 df-rab 3075 df-v 3433 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-nul 4263 df-if 4466 df-sn 4568 df-pr 4570 df-op 4574 df-br 5080 df-opab 5142 df-xp 5595 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-ec 8481 df-coss 36531 |
This theorem is referenced by: eldmqs1cossres 36765 |
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